Recall that W satisfies the pentagon equation and the coaction axiom (2.35). In a more familiar form, we see that W=W21:=W∈ M(A⊗A)satisfies
(1⊗)W=W12 W13 . (6.33)
Hence, if we treat Was an element in M(B(H)⊗A)instead, we obtain the left regular corepresentation
:H−→H⊗M(A)
f → W(f ⊗1) (6.34)
which satisfies
(1⊗)◦=(⊗1)◦. (6.35)
This corepresentation picture is useful when we study the corepresentation of the quantum double. Therefore, we introduce the transformation fromHtoHcorepso that Aacts canonically. Similarly, by consideringW , we also describe the transformation so thatAacts canonically onH.
Definition 6.12 We define the transformationTco:H−→Hcorepby Tco: f(s,t)→ f
−s,s−t+i Q 2
Gb
Q
2 −i s+i t −1
, (6.36) Tco−1: f(s,t)→ f
−s,−t−s+i Q 2
Gb(−i t). (6.37)
Similarly, we define the transformationTco:H−→Hcorepby Tco: f(s,t)→ f
−s,s−t+i Q 2
Gb
Q
2 +i s−i t
e2πi s
s−t+i Q2
,
(6.38) Tco−1: f(s,t)→ f
−s,−t−s+i Q 2
e2πi stGb(Q+i t)−1. (6.39) Proposition 6.13 Under the transformationTco, the action ofAonHcorepis given by
A=e−2πbs, B =e2πbps. (6.40) Under the transformationTco, the action ofAonHcorepis given by
A=e2πbs, B=e2πbps. (6.41)
Furthermore, the L2measure onHcorepandHcorepbecomes the standard Lebesgue measure.
Proof It suffices to see that the transformations can be written as Tco=(t →t−s)◦(−s,−t)◦
followed by applying the transformation rules.
This choice allows us to reproduce the representation from the pairing betweenA andAgiven by the corepresentation associated with the multiplicative unitary.
Proposition 6.14 The left regular corepresentation associated with WonAis given by Remark 6.15 Note that the left regular corepresentation is related to the left “funda-mental” representation represen-tation of the quantum plane algebraBq=A∗qis given by the canonical action
X=e2πbs, Y =e2πb(ps+s). (6.47) Similarly, we can define the action of its modular double counterpart
X =e2πb−1s, Y=e2πb−1(ps+s), (6.48)
so that it extends to a representation of the modular doubleBqq. Under a unitary transform by multiplication by eπi s2 on Hcorep, the action becomes the canonical action ofBqq.
Proof It follows directly from Proposition5.11sinceλis the identity map; hence, the elements ofAare sent to the corresponding actions. Now, the formula follows from
the definition X =A−1,Y =qBA−1.
Similarly, the corepresentation associated withWis given by f(s)→
f(s+τ)Gb(−iτ)Bi b−1τAi b−1sdτ. (6.49)
For completeness, by composingTcowithT−1, we obtain the transformationS : Hrep−→Hcorep
Proposition 6.17 The transformationS :Hrep−→Hcorepis given by S: f(s,t)→
R
f(α−s, α−t)Gb
Q
2 −i s+i t −1
e2πi(α−s)sdα, (6.50) S−1: f(s,t)→
R
f(α−s, α−t)Gb
Q
2 +i s−i t
e−2πi(α−s)sdα.
(6.51) Similarly, we define the corresponding transformation forAusingS:Hrep −→
Hcorepby
S: f(s,t)→
R
f(α−s, α−t)Gb
Q
2 +i s−i t
e−2πi(α−s)sdα, (6.52) S−1: f(s,t)→
R
f(α−s, α−t)Gb
Q
2 −i s+i t −1
e2πi(α−s)sdα. (6.53)
7 The quantum double construction
7.1 Definitions
In this section, we will describe the quantum double group construction given by [30] (see also [34]) associated with the quantum plane and show that the object we obtain is exactly the quantum “semigroup” G Lq+(2,R), also called the split quantum
Minkowski spacetime, which is a generalization of the compact Minkowski spacetime introduced in [8,9].
Definition 7.1 We define the split quantum Minkowski Spacetime M+q(R)as the Hopf *-algebra generated by positive self-adjoint operators zi j,i,j ∈ {1,2}such that the following relations hold:
[z11,z12]=0, [z21,z22]=0, [z11,z22]= [z12,z21],
z11z21 =q2z21z11, z12z22 =q2z22z12, z12z21 =q2z21z12, and the coproduct is given by
(z11)=z11⊗z11+z12⊗z21, (7.1) (z12)=z11⊗z12+z12⊗z22, (7.2) (z21)=z21⊗z11+z22⊗z21, (7.3) (z22)=z21⊗z12+z22⊗z22. (7.4) It can also be realized as G L+q(2,R)in matrix form:
z11 z12
z21 z22
(7.5) so that the coproduct is simply given by
z11 z12
z21 z22
=
z11 z12
z21 z22
⊗
z11 z12
z21 z22
. (7.6)
The quantum determinant N is the positive self-adjoint operator defined by N =z11z22−z12z21 =z22z11−z21z12, (7.7) and we have
N z11 =z11N, N z12 =q−2z12N, N z21=q2z21N, N z22=z22N. (7.8) Proposition 7.2 There is a projection mapP :G L+q(2,R)−→S L+q(2,R)given by
z11 z12
z21 z22
→ a b
c d
:=
N−1/2z11 q−1/2N−1/2z12
q1/2N−1/2z21 N−1/2z22
, (7.9)
where a,b,c,d satisfies the usual relations for S Lq(2,R):
ab=qba, ac=qca, ad=qda, bd =qdb, cd =qdc,
bc=cb, ad−qbc=da−q−1cb=1. (7.10) Proposition 7.3 There is a Gauss decomposition for G L+q(2,R)given by
z11 z12
z21 z22
= A 0
B 1
1 B 0 A
=
A AB B BB+A
, (7.11)
where A,B,A,B are positive operators so that{A,B}commutes with{A,B}, with A B=q2B A and AB=q−2BA. Furthermore, we have N =AA.
Now, we will describe the quantum double group construction and show that the result is precisely G L+q(2,R)together with the above Gauss decomposition.
Definition 7.4 The quantum double groupD(A)is the Hopf algebra where as an algebraD(A)A⊗A∗op=A⊗Awith the usual tensor product algebra structure and with coproduct given by
m(x⊗x)=(1⊗σm⊗1)((x)⊗(x)), (7.12) whereσis the permutation of the tensor product, and m:M(A⊗A) −→M(A⊗A) is called the matching, defined by
m(x⊗x)=W(x⊗x)W∗, (7.13)
with W ∈ M(A⊗A) the multiplicative unitary defined in (4.61).
Hence, a general element inD(A)can be written as
f(s1,t1)g(s2,t2)Ai b−1s1Bi b−1t1Bi b−1t2Ai b−1s2ds1dt1ds2dt2 (7.14) or simply f(s1,t1)g(s2,t2). For simplicity, we write A:=A⊗1, A:=1⊗A and so on, and we will use this notation in the remaining of the paper.
Proposition 7.5 [30, Thm 4.1]m is coassociative, so it indeed defines a coproduct onD(A).
Proposition 7.6 [30, Thm 4.2] The Haar functional h onD(A)defined by
h=hL⊗hR (7.15)
is both left and right invariant:
(h⊗1⊗1)m(x⊗x)=h(x⊗x)(1⊗1), (7.16) (1⊗1⊗h)m(x⊗x)=h(x⊗x)(1⊗1). (7.17)
In particular, the GNS representation ofD(A)is given bym :=⊗RonH⊗H.
Proof Although the theorem in [30] applies only to compact quantum groups, the calculations using the graphical method there can be adapted in this setting without any changes, since it only depends on the invariances for hL,hR and the relations between the matching m and the coproducts ofAandA.
Remark 7.7 The Haar functional on a general element f(s1,t1)g(s2,t2)∈H⊗His thus given by
h(f ⊗g)= f(0,i Q)g(−i Q,i Q)e−πi Q2. (7.18) If we parametrize the element instead as
f ⊗g := f(s1,t1)g(s2,t2)Ai b−1s1Bi b−1t1Xi b−1s2Yi b−1t2ds1dt1ds2dt2, (7.19) where X= A−1,Y =qBA−1, then it takes a more symmetric form
h(f ⊗g)= f(0,i Q)g(0,i Q), (7.20) which means it only depends on the element BY . According to the Gauss decompo-sition, this is precisely
B(qBA−1)=q BB A A−1A−1=q z21z12N−1, (7.21) which corresponds under the projection to S L+q(2,R)the hyperbolic elementζ :=bc that is crucial in the study of the SUq(2)and SUq(1,1)case in [27,28].
Theorem 7.8 By the Gauss decomposition, the Hopf algebra G L+q(2,R)can be nat-urally put into the C∗-algebraic setting, so that it is identified with the quantum double D(A). Furthermore, the coproduct onD(A)induces the same coproduct on the gen-erators zi j.
Proof By the Gauss decomposition, there is a one-to-one correspondence between the generators. Explicitly, the inverse is given by
A=z11, B =z21, B =z12z−111, A=N z−111.
We have to show that the coproduct is the same. The following calculations are very
similar to the one given in [34].
Lemma 7.9 We have the following commutation relationships between W andD(A):
W(A⊗1)W∗= A⊗1+B⊗B, W(A⊗A)W∗= A⊗A,
W(A⊗B)W∗=1⊗B, W(B⊗1)W∗= B⊗A,
W∗(1⊗A)W =1⊗A+B⊗B.
Proof These follow from the summation properties (3.22),(3.23) for gb, as well as the commutation relations (2.46),(2.47) for the exponentials.
Now, we proceed to the calculation of the coproduct on the generators:
(z11)=(A⊗1)
= A⊗σm(A⊗1)⊗1
=(A⊗1)⊗(A⊗1)+(A⊗B)⊗(B⊗1)
=z11⊗z11+z12⊗z21, (z12)=(A⊗B)
= A⊗σm(A⊗1)⊗B+A⊗σm(A⊗B)⊗A
= A⊗1⊗A⊗B+A⊗B⊗B⊗B+A⊗B⊗1⊗A
=(A⊗1)⊗(A⊗B)+(A⊗B)⊗(B⊗B+1⊗A)
=z11⊗z12+z12⊗z22, (z21)=(B⊗1)
=B⊗σm(A⊗1)⊗1+1⊗σm(B⊗1)⊗1
=B⊗1⊗A⊗1+B⊗B⊗B⊗1+1⊗A⊗B⊗1
=(B⊗1)⊗(A⊗1)+(B⊗B+1⊗A)⊗(B⊗1)
=z21⊗z11+z22⊗z21, (z22)=(B⊗B+1⊗A)
=B⊗σm(A⊗1)⊗B+1⊗σm(B⊗1)⊗B+B⊗σm(A⊗B)⊗A +1⊗σm(B⊗B+1⊗A)⊗A.
Since W∗(1⊗A)W =1⊗A+B⊗B, we have
σm(B⊗B+1⊗A)=A⊗1. Hence,
(z22)=B⊗1⊗A⊗B+B⊗B⊗B⊗B+1⊗A⊗B⊗B +B⊗B⊗1⊗A+1⊗A⊗1⊗A
=(B⊗1)⊗(A⊗B)+(B⊗B+1⊗A)⊗(B⊗B+1⊗A)
=z21⊗z12+z22⊗z22.
Finally, let us also derive the coproduct for the determinant N : (N)=(A⊗A)
= A⊗σm(A⊗A)⊗A
= A⊗A⊗A⊗A
=N⊗N.